Solve The Inequality: X 3 − X 2 \textgreater 1 \frac{x}{3} - \frac{x}{2} \ \textgreater \ 1 3 X − 2 X \textgreater 1

Mar 11, 2025 by ADMIN 123 views

Solving Inequalities: A Step-by-Step Guide to $\frac{x}{3} - \frac{x}{2} \ \textgreater \ 1$

Inequalities are mathematical expressions that compare two values, often with a greater-than or less-than symbol. Solving inequalities involves finding the values of the variable that make the inequality true. In this article, we will focus on solving the inequality $\frac{x}{3} - \frac{x}{2} \ \textgreater \ 1$ . We will break down the solution into manageable steps, using algebraic manipulations and logical reasoning to arrive at the final answer.

Step 1: Write Down the Inequality

The given inequality is $\frac{x}{3} - \frac{x}{2} \ \textgreater \ 1$ . Our goal is to isolate the variable $x$ and find the values that satisfy the inequality.

Step 2: Combine Like Terms

To simplify the inequality, we can combine the like terms on the left-hand side. We have two fractions with different denominators, so we need to find a common denominator. The least common multiple of 3 and 2 is 6, so we can rewrite the inequality as:

\frac{2x}{6} - \frac{3x}{6} \ \textgreater \ 1

Now we can combine the fractions:

-\frac{x}{6} \ \textgreater \ 1

Step 3: Multiply Both Sides by -1

To get rid of the negative sign on the left-hand side, we can multiply both sides of the inequality by -1. However, when we multiply an inequality by a negative number, we need to reverse the direction of the inequality sign. So, we get:

\frac{x}{6} \ \textless \ -1

Step 4: Multiply Both Sides by 6

To isolate the variable $x$ , we can multiply both sides of the inequality by 6. This will eliminate the fraction on the left-hand side:

x \ \textless \ -6

Step 5: Write the Final Answer

The final answer is $x \ \textless \ -6$ . This means that any value of $x$ that is less than -6 will satisfy the original inequality.

Solving inequalities involves a series of algebraic manipulations and logical reasoning. By following the steps outlined in this article, we can solve the inequality $\frac{x}{3} - \frac{x}{2} \ \textgreater \ 1$ and find the values of $x$ that satisfy the inequality. Remember to always check your work and verify the solution by plugging in values to ensure that the inequality holds true.

Common Mistakes to Avoid

When solving inequalities, it's easy to make mistakes. Here are some common pitfalls to avoid:

Not checking the direction of the inequality sign: When multiplying an inequality by a negative number, make sure to reverse the direction of the inequality sign.
Not simplifying the inequality: Make sure to combine like terms and simplify the inequality as much as possible.
Not verifying the solution: Always check your work by plugging in values to ensure that the inequality holds true.

Real-World Applications

Solving inequalities has many real-world applications. Here are a few examples:

Finance: In finance, inequalities are used to model investment returns and risk. For example, an investor may want to know the minimum return on investment required to achieve a certain level of wealth.
Engineering: In engineering, inequalities are used to model physical systems and optimize performance. For example, an engineer may want to design a system that meets certain performance criteria, such as maximum speed or minimum cost.
Science: In science, inequalities are used to model physical phenomena and make predictions. For example, a scientist may want to model the behavior of a complex system, such as a population growth model.

Solving inequalities is an essential skill in mathematics and has many real-world applications. By following the steps outlined in this article, you can solve inequalities and make informed decisions in a variety of fields. Remember to always check your work and verify the solution by plugging in values to ensure that the inequality holds true.
Solving Inequalities: A Q&A Guide

In our previous article, we explored the steps to solve the inequality $\frac{x}{3} - \frac{x}{2} \ \textgreater \ 1$ . We broke down the solution into manageable steps, using algebraic manipulations and logical reasoning to arrive at the final answer. In this article, we will address some common questions and concerns that readers may have when solving inequalities.

Q: What is the difference between an inequality and an equation?

A: An equation is a mathematical statement that says two expressions are equal, while an inequality is a mathematical statement that says one expression is greater than or less than another expression.

Q: How do I know which direction to use when solving an inequality?

A: When solving an inequality, you need to determine the direction of the inequality sign. If you multiply or divide both sides of the inequality by a negative number, you need to reverse the direction of the inequality sign.

Q: What is the least common multiple (LCM) and how do I find it?

A: The least common multiple (LCM) is the smallest multiple that two or more numbers have in common. To find the LCM, you can list the multiples of each number and find the smallest multiple that appears in both lists.

Q: How do I simplify an inequality with fractions?

A: To simplify an inequality with fractions, you can multiply both sides of the inequality by the least common multiple (LCM) of the denominators. This will eliminate the fractions and make it easier to solve the inequality.

Q: What is the difference between a linear inequality and a quadratic inequality?

A: A linear inequality is an inequality that can be written in the form $ax + b \ \textgreater \ c$ , where $a$ , $b$ , and $c$ are constants. A quadratic inequality is an inequality that can be written in the form $ax^2 + bx + c \ \textgreater \ d$ , where $a$ , $b$ , $c$ , and $d$ are constants.

Q: How do I solve a quadratic inequality?

A: To solve a quadratic inequality, you can use the following steps:

Factor the quadratic expression, if possible.
Set each factor equal to zero and solve for $x$ .
Use a number line or a graph to determine the intervals where the inequality is true.

Q: What is the difference between a strict inequality and a non-strict inequality?

A: A strict inequality is an inequality that uses a strict greater-than or less-than symbol, such as $x \ \textgreater \ 2$ . A non-strict inequality is an inequality that uses a greater-than or equal-to or less-than or equal-to symbol, such as $x \ \textgreater \ 2$ or $x \ \textless \ 2$ .

Q: How do I graph an inequality on a number line?

A: To graph an inequality on a number line, you can use the following steps:

Draw a number line and mark the point that corresponds to the value of $x$ .
Use a closed circle to indicate the point that corresponds to the value of $x$ if the inequality is non-strict.
Use an open circle to indicate the point that corresponds to the value of $x$ if the inequality is strict.
Shade the region to the left or right of the point, depending on the direction of the inequality sign.

Solving inequalities can be a challenging task, but with practice and patience, you can become proficient in solving them. Remember to always check your work and verify the solution by plugging in values to ensure that the inequality holds true. If you have any further questions or concerns, feel free to ask.

Common Mistakes to Avoid

When solving inequalities, it's easy to make mistakes. Here are some common pitfalls to avoid:

Not checking the direction of the inequality sign: When multiplying an inequality by a negative number, make sure to reverse the direction of the inequality sign.
Not simplifying the inequality: Make sure to combine like terms and simplify the inequality as much as possible.
Not verifying the solution: Always check your work by plugging in values to ensure that the inequality holds true.

Real-World Applications

Solving inequalities has many real-world applications. Here are a few examples:

Finance: In finance, inequalities are used to model investment returns and risk. For example, an investor may want to know the minimum return on investment required to achieve a certain level of wealth.
Engineering: In engineering, inequalities are used to model physical systems and optimize performance. For example, an engineer may want to design a system that meets certain performance criteria, such as maximum speed or minimum cost.
Science: In science, inequalities are used to model physical phenomena and make predictions. For example, a scientist may want to model the behavior of a complex system, such as a population growth model.